The \(u\)-gamma functions and the \(u\)-exponentials of one and several complex variables are introduced and deformations of general hypergeometric functions are then studied by them. In the case of functions of one variable, the infinite product representations of the gamma functions \(\Gamma_u\) are obtained and their asymptotic behavior is investigated. The conditions ensuring the convergence of the formal series \(\exp_u x\) are discussed. The definitions of \(\Gamma_u\), \(\exp_ux\) and \(\partial_u\) are generalized to the many-dimensional case. The reduced \(u\)-hypergeometric functions associated with a lattice \(\Lambda\subset \mathbb{C}^N\), and a vector \(\lambda\in \mathbb{C}^N\) are introduced. It is shown that all the classical (Pochhammer, Appell, Lauricella, etc.) hypergeometric functions of one or several variables are reduced hypergeometric functions associated with appropriate \(\Lambda\) and \(\lambda\).